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A336590 Numbers k such that k/A008834(k) is squarefree, where A008834(k) is the largest cube dividing k.

%I A336590 #10 Jul 28 2020 10:31:36
%S A336590 1,2,3,5,6,7,8,10,11,13,14,15,16,17,19,21,22,23,24,26,27,29,30,31,33,
%T A336590 34,35,37,38,39,40,41,42,43,46,47,48,51,53,54,55,56,57,58,59,61,62,64,
%U A336590 65,66,67,69,70,71,73,74,77,78,79,80,81,82,83,85,86,87,88
%N A336590 Numbers k such that k/A008834(k) is squarefree, where A008834(k) is the largest cube dividing k.
%C A336590 Numbers such that none of the exponents in their prime factorization is of the form 3*m + 2.
%C A336590 Cohen (1962) proved that for a given number k >= 2 the asymptotic density of numbers whose exponents in their prime factorization are of the forms k*m or k*m + 1 only is zeta(k)/zeta(2). In this sequence k = 3, and therefore its asymptotic density is zeta(3)/zeta(2) = 6*zeta(3)/Pi^2 = 0.7307629694... (A253905).
%C A336590 Also, numbers k whose number of divisors, A000005(k), is not divisible by 3, i.e., complement of A059269.
%H A336590 Amiram Eldar, <a href="/A336590/b336590.txt">Table of n, a(n) for n = 1..10000</a>
%H A336590 Eckford Cohen, <a href="https://projecteuclid.org/euclid.pjm/1103036708">Arithmetical notes. III. Certain equally distributed sets of integers</a>, Pacific Journal of Mathematics, No. 12, Vol. 1 (1962), pp. 77-84.
%H A336590 Eckford Cohen, <a href="https://eudml.org/doc/140760">Arithmetical Notes, XIII. A Sequal to Note IV</a>, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.
%H A336590 L. G. Sathe, <a href="https://www.jstor.org/stable/2371953">On a congruence property of the divisor function</a>, American Journal of Mathematics, Vol. 67, No. 3 (1945), pp. 397-406.
%e A336590 6 is a term since 6 = 2^1 * 3^1 and 1 is not of the form 3*m + 2.
%e A336590 9 is not a term since 9 = 3^2 and 2 is of the form 3*m + 2.
%t A336590 Select[Range[100], Max[Mod[FactorInteger[#][[;;,2]], 3]] < 2 &]
%Y A336590 Cf. A000005, A002117, A005117, A008834, A013661, A059269, A253905.
%Y A336590 Union of A211337 and A211338.
%K A336590 nonn
%O A336590 1,2
%A A336590 _Amiram Eldar_, Jul 26 2020